Research on the effect of the heavy nuclei amount on the temperature reactivity coefficient in a small modular molten salt reactor

Research on the effect of the heavy nuclei amount on the temperature reactivity coefficient in a small modular molten salt reactor a b first-author a corresp zhuguifeng@sinap.ac.cn zhuguifeng@sinap.ac.cn zhuguifeng@sinap.ac.cn a a corresp Corresponding author, e-mail address: yuxiaohan@sinap.ac.cn; Corresponding author, e-mail address: yuxiaohan@sinap.ac.cn; yuxiaohan@sinap.ac.cn a Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China University of Chinese Academy of Sciences, Beijing 100049, China Small modular thorium-based graphite-moderated molten salt reactors (smTMSRs), which combine the advantages of small modular reactors (SMRs) and molten salt reactors (MSRs), are regarded as a wise development path to speed deployment time. In an smTMSR, low enriched uranium (LEU) and thorium fuel are used in once-through mode, which makes a marked difference in their neutronic properties compared with the case when a conventional molten salt breeder reactor is used. This study investigated the temperature reactivity coefficient (TRC) in an smTMSR, which is mainly affected by the molten salt volume fraction (VF) and the heavy nuclei concentration in the fuel salt (HN). The four-factor formula method and the reaction rate method were used to indicate the reasons for the TRC change, including the fuel density effect, the fuel Doppler effect, and the graphite thermal scattering effect. The results indicate that only the fuel density has a positive effect on the TRC in the undermoderated region. Thermal scattering from both salt and graphite have a significant negative influence on the TRC in the overmoderated region. The maximal effective multiplication factor, which shows the highest fuel utilization, is located at 10% VF and 12 mol% HN, and is still located in the negative TRC region. In addition, on increasing the heavy nuclei amount from 2 mol% HN to 12 mol% HN (VF=10%), the total TRC undergoes an obvious change from -11 pcm/K to -3 pcm/K, which implies that the change in the HN caused by the fuel feed online should be small to avoid potential trouble in the reactivity control scheme. Molten salt reactor Temperature reactivity coefficient Heavy nuclei amount 1. Introduction 1 Change in the temperature of a nuclear reactor leads to change in the reactivity of the system, which is described in terms of the temperature reactivity coefficient (TRC) [ 1 1 ]. The TRC of a reactor must be negative to guarantee its self-control ability, and thus plays a crucial role in reactor safety and operation. Molten salt reactors (MSRs) [ 2 2 - 3 3 2-3 ] have special TRCs due to not only their high temperature materials, but also to their liquid nuclear fuel. In an MSR, the total reactivity feedback of temperature can be split into effects due to the graphite and those due to the fuel salt. The latter can be further split into the density effect and the Doppler effect. At present, research into the TRC of MSRs is mostly based on 233 U- 232 Th fuel in conventional molten salt breeder reactors (MSBRs) [ 4 4 ]. Many studies have shown that the TRC of the MSBR can be positive when the molten salt fraction increases, due to the positive graphite temperature reactivity coefficient (GTRC) [ 5 5 - 10 10 5-10 ]. One of the positive contributions always comes from the graphite neutron scattering effect, which increases the thermal fission rate of 233 U when the temperature of graphite increases. Another positive contribution may come from the density effect of the salt in the under-moderated region. Therefore, many design optimizations of MSBRs have been performed to obtain a negative total TRC. Recently, MSRs using low enriched uranium (LEU) with a once-through fuel cycle have aroused great interest around the world. During operation, LEU is added to the online reactor to compensate for the reactivity loss, and no chemical reprocessing is carried out. This kind of fuel cycle is derived from the denatured molten salt reactor (DMSR) [ 11 11 ], whose purpose is to minimize nuclear proliferation risk, and is currently implemented by ThorCon [ 12 12 ] and IMSR [ 13 13 ] for the rapid deployment of MSRs due to fuel availability and less technical challenges. Also, the LEU fuel cycle is adopted in the small modular thorium-based molten salt reactor (smTMSR), as the second stage of the thorium-based molten salt reactor (TMSR) nuclear energy system project launched by the Chinese Academy of Sciences (CAS) [ 9 9 , 14 14 , 15 15 ]. Two differences can be found in the LEU fuel cycle when compared with the fuel cycle in conventional MSBRs. One is that the fuel is 235 U- 238 U or 235 U- 238 U- 232 Th instead of 233 U- 232 Th; these different fission cross sections will bring about different Doppler effects and thermal scattering effects, especially in terms of the GTRC. Another is that the amount of heavy nuclei will escalate during the burnup life, which will contribute to a significant change in the neutron spectrum as well as the reaction rates of fuel, graphite, and other materials, leading to the change in the TRC [ 16 16 , 17 17 ]. The TRCs in the LEU fuel cycle were calculated for some designs of MSR[ 8 8 , 9 9 , 15 15 , 16 16 , 18 18 ] to show their inherent safety, while less systematical research and the mechanism of the effect on the TRC have been carried out. In this paper, the fuel density coefficient, the fuel Doppler coefficient, and the GTRC of the smTMSR with different heavy nuclei amounts were analyzed. The four-factor formula method and reaction rate analysis were also employed to explain the changes in the TRC in the current study. Sec. 2 introduces the model of the smTMSR and Sec. 3 introduces the research methods. Sec. 4 presents the results and analysis. The conclusion is given in Sec. 5. 2. Calculation model 1 The smTMSR is a 400-MWth thermal reactor based on modular technology. It can be applied in coastal areas such as the seaside, islands and offshore platforms, and it can also be used in inland, mountainous, and mining areas. A preliminary nuclear design of the smTMSR [ 19 19 ] is shown in Fig. 1 Fig. 1 and Table 1 Table 1 . The core consists of hexagonal prism graphite blocks and the molten salt channels. Large graphite blocks are adopted to enhance the space self-shielding effect of 238 U. 30 cm width graphite reflectors are added to reduce fast neutron irradiation of the Hastelloy reactor vessel. The core diameter is set as 4.4 m to improve the fuel utilization and lower the power density to extend the graphite irradiation life. The ratios of fuel composition are mole ratios. 1 1 Parameter 1 1 Value 1 1 Power (MWth) 1 1 400 1 1 Inlet/Outlet temperature (℃) 1 1 600/700 1 1 Hexagonal graphite component pitch (cm) 1 1 26 1 1 Active region height/diameter (m) 1 1 4.4 /4.4 1 1 Graphite reflector thickness (m) 1 1 0.3 1 1 Hastelloy vessel thickness (cm) 1 1 2 1 1 Fuel salt composition 1 1 LiF:BeF 2 =73.83:18.99 1 1 1 1 235 UF 4 : 238 UF 4 :ThF 4 =0.17:0.68:6.33 1 1 7 Li enrichment (mol%) 1 1 99.995 1 1 900 K, Fuel salt density (g/cm 3 ) [ 21 21 ] (HN=2/4/6/8/10 mol%) 1 1 2.195/2.439/2.665/2.873/3.068 1 1 1000 K, Fuel salt density (g/cm 3 ) (HN=2/4/6/8/10 mol%) 1 1 2.156/2.397/2.619/2.825/3.017 Changes in the heavy nuclei amount are mainly reflected in the mole concentration of heavy nuclei fuel in the molten salt (HN) and the volume fraction of molten salt in the active core (VF). In order to extensively investigate the effect of the heavy nuclei amount on the TRC, we varied the HN from 2 mol% to 12 mol% (2, 4, 6, 8, 10, and 12 mol%) and the VF from 5% to 25% (5, 10, 15, 20, and 25%) at the beginning of the lifetime, which meant that a total of 25 different cores were calculated. The TRC was calculated from difference of 100 K (from 900 K to 1000 K). The number of particles used in MCNP5 [ 20 20 ] was 500,000, and the number of total active cycles was 350. The statistical standard deviation of k eff reached 0.00005. The ENDF/B-VII library was used in this calculation. 3. Four-factor formula method 1 One of the TRC analysis methods [ 22 22 - 24 24 22-24 ] used was the four-factor formula method. In this method, <math display="block" id="M1"><mrow><msub><mi>k</mi><mi>∞</mi></msub><mo>=</mo><mi>f</mi><mi>η</mi><mi>p</mi><mi>ε</mi><mo> </mo><mo>,</mo></mrow></math> where <math id="M2"><mrow><msub><mi>k</mi><mi>∞</mi></msub></mrow></math> is the infinite multiplication factor, f is the thermal utilization factor, <math id="M3"><mrow><mtext> </mtext><mi>η</mi></mrow></math> is the thermal fission factor, p is the resonance escape probability, and <math id="M4"><mi>ε</mi></math> is the fast fission factor. The formula for each factor is: [ 25 25 - 27 27 25-27 ] <math display="block" id="M5"><mrow><mi>ε</mi><mo>=</mo><mfrac><mrow><mi>ν</mi><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>f</mi><mo>,</mo><mn>1</mn></mrow><mi>f</mi></msubsup><msub><mi>ϕ</mi><mn>1</mn></msub><mo>+</mo><mi>ν</mi><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>f</mi><mo>,</mo><mn>2</mn></mrow><mi>f</mi></msubsup><msub><mi>ϕ</mi><mn>2</mn></msub></mrow><mrow><mi>ν</mi><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>f</mi><mo>,</mo><mn>2</mn></mrow><mi>f</mi></msubsup><msub><mi>ϕ</mi><mn>2</mn></msub></mrow></mfrac><mo> </mo><mo>,</mo></mrow></math> <math display="block" id="M6"><mrow><mi>p</mi><mo>=</mo><mfrac><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>a</mi><mo>,</mo><mn>2</mn></mrow></msub><msub><mi>ϕ</mi><mn>2</mn></msub></mrow><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>a</mi><mo>,</mo><mn>1</mn></mrow></msub><msub><mi>ϕ</mi><mn>1</mn></msub><mo>+</mo><msub><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>a</mi><mo>,</mo><mn>2</mn></mrow></msub><msub><mi>ϕ</mi><mn>2</mn></msub></mrow></mfrac><mtext> </mtext><mo>,</mo></mrow></math> <math display="block" id="M7"><mrow><mi>η</mi><mo>=</mo><mfrac><mrow><mi>ν</mi><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>f</mi><mo>,</mo><mn>2</mn></mrow><mi>f</mi></msubsup><msub><mi>ϕ</mi><mn>2</mn></msub></mrow><mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>a</mi><mo>,</mo><mn>2</mn></mrow><mi>f</mi></msubsup><msub><mi>ϕ</mi><mn>2</mn></msub></mrow></mfrac><mo> </mo><mo>,</mo></mrow></math> <math display="block" id="M8"><mrow><mi>f</mi><mo>=</mo><mfrac><mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>a</mi><mo>,</mo><mn>2</mn></mrow><mi>f</mi></msubsup><msub><mi>ϕ</mi><mn>2</mn></msub></mrow><mrow><msub><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>a</mi><mo>,</mo><mn>2</mn></mrow></msub><msub><mi>ϕ</mi><mn>2</mn></msub><mo> </mo></mrow></mfrac><mo> </mo><mo>,</mo></mrow></math> where <math id="M9"><mi>ν</mi></math> is the average fission neutrons produced for each fission, <math id="M10"><mrow><mo> </mo><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>f</mi><mo>,</mo><mn>1</mn></mrow><mi>f</mi></munderover><msub><mi>ϕ</mi><mn>1</mn></msub></mrow></math> is the fuel fast fission rate, <math id="M11"><mrow><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>f</mi><mo>,</mo><mn>2</mn></mrow><mi>f</mi></munderover><msub><mi>ϕ</mi><mn>2</mn></msub></mrow></math> is the fuel thermal fission rate, <math id="M12"><mrow><munder><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>a</mi><mo>,</mo><mn>2</mn></mrow></munder><msub><mi>ϕ</mi><mn>2</mn></msub></mrow></math> is the thermal absorption rate, <math id="M13"><mrow><munder><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>a</mi><mo>,</mo><mn>1</mn></mrow></munder><msub><mi>ϕ</mi><mn>1</mn></msub></mrow></math> is the fast absorption rate, and <math id="M14"><mrow><mo> </mo><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>a</mi><mo>,</mo><mn>2</mn></mrow><mi>f</mi></munderover><msub><mi>ϕ</mi><mn>2</mn></msub></mrow></math> is the fuel thermal absorption rate. The thermal energy group ranges from 0 to 0.625 eV, and the fast energy group ranges from 0.625 eV to 20 MeV. According to the definition of the effective multiplication factor, <math display="block" id="M15"><mrow><msub><mi>k</mi><mrow><mtext>eff</mtext></mrow></msub><mo>=</mo><mi>f</mi><mi>η</mi><mi>p</mi><mi>ε</mi><mtext>Λ </mtext><mo>,</mo></mrow></math> where <math id="M16"><mtext>Λ</mtext></math> is the non-leakage probability. In this study, Λ is the ratio of the core active area absorption rate to the total absorption rate. The reaction rate of fission or absorption can be obtained using MCNP. Therefore, we can determine the values of <math id="M17"><mrow><mi>f</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>ε</mi></mrow></math> and <math id="M18"><mtext>Λ</mtext></math> . The reactivity change per unit temperature change is the TRC, which can be expressed as: <math display="block" id="M19"><mrow><msub><mi>α</mi><mtext>T</mtext></msub><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>ρ</mi></mrow><mrow><mo>∂</mo><mi>T</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>k</mi><mrow><mtext>eff</mtext></mrow></msub></mrow></mfrac><mfrac><mrow><mo>∂</mo><msub><mi>k</mi><mrow><mtext>eff</mtext></mrow></msub></mrow><mrow><mo>∂</mo><mi>T</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>k</mi><mrow><mtext>eff</mtext></mrow></msub><mo>−</mo><mn>1</mn></mrow><mrow><msubsup><mi>k</mi><mrow><mtext>eff</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><mfrac><mrow><mo>∂</mo><msub><mi>k</mi><mrow><mtext>eff</mtext></mrow></msub></mrow><mrow><mo>∂</mo><mi>T</mi></mrow></mfrac><mo> </mo><mo>.</mo></mrow></math> The reactivity changes for the individual factors are defined as: [ 28 28 ] <math display="block" id="M20"><mrow><msubsup><mi>α</mi><mtext>T</mtext><mi>f</mi></msubsup><mo>=</mo><mfrac><mrow><mi>ε</mi><mi>p</mi><mi>η</mi><mtext>ΛΔ</mtext><mi>f</mi></mrow><mrow><mtext>Δ</mtext><mi>T</mi><msubsup><mi>k</mi><mrow><mtext>eff</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><mo>,</mo><mtext> </mtext><msubsup><mi>α</mi><mtext>T</mtext><mi>η</mi></msubsup><mo>=</mo><mfrac><mrow><mi>ε</mi><mi>p</mi><mi>f</mi><mtext>ΛΔ</mtext><mi>η</mi></mrow><mrow><mtext>Δ</mtext><mi>T</mi><msubsup><mi>k</mi><mrow><mtext>eff</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><mo>,</mo><mtext> </mtext><msubsup><mi>α</mi><mtext>T</mtext><mi>p</mi></msubsup><mo>=</mo><mfrac><mrow><mi>ε</mi><mi>p</mi><mi>f</mi><mtext>ΛΔ</mtext><mi>p</mi></mrow><mrow><mtext>Δ</mtext><mi>T</mi><msubsup><mi>k</mi><mrow><mtext>eff</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><mo>,</mo><mtext> </mtext><msubsup><mi>α</mi><mtext>T</mtext><mi>ε</mi></msubsup><mo>=</mo><mfrac><mrow><mi>p</mi><mi>η</mi><mi>f</mi><mtext>ΛΔ</mtext><mi>ε</mi></mrow><mrow><mtext>Δ</mtext><mi>T</mi><msubsup><mi>k</mi><mrow><mtext>eff</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><mo>,</mo><mtext> </mtext><msubsup><mi>α</mi><mtext>T</mtext><mtext>Λ</mtext></msubsup><mo>=</mo><mfrac><mrow><mi>ε</mi><mi>p</mi><mi>η</mi><mi>f</mi><mtext>ΔΛ</mtext></mrow><mrow><mtext>Δ</mtext><mi>T</mi><msubsup><mi>k</mi><mrow><mtext>eff</mtext></mrow><mn>2</mn></msubsup></mrow></mfrac><mo>.</mo></mrow></math> Through combining Eqs. (6), (7) and (8), the sum is obtained as: <math display="block" id="M21"><mrow><msub><mi>α</mi><mtext>T</mtext></msub><mo>=</mo><msubsup><mi>α</mi><mtext>T</mtext><mi>f</mi></msubsup><mo>+</mo><msubsup><mi>α</mi><mtext>T</mtext><mi>η</mi></msubsup><mo>+</mo><msubsup><mi>α</mi><mtext>T</mtext><mi>p</mi></msubsup><mo>+</mo><msubsup><mi>α</mi><mtext>T</mtext><mi>ε</mi></msubsup><mo>+</mo><msubsup><mi>α</mi><mtext>T</mtext><mtext>Λ</mtext></msubsup><mo> </mo><mo>.</mo></mrow></math> According to Eq. (9), the TRC can be approximately decomposed into five parts. In this way, the contribution of each factor to the TRC can be studied, and the main factors influencing the TRC can be obtained. The TRC can be decomposed into the fuel TRC (FTRC) and the GTRC [ 6 6 ]. Based on the density change in the molten salt, the FTRC is divided into the TRC caused by the density effect (the fuel density coefficient) and the TRC resulting from the Doppler effect (the fuel Doppler coefficient). The GTRC, the fuel density coefficient, and the fuel Doppler coefficient can be calculated by the definition of the TRC. Therefore, the total TRC of the smTMSR can be expressed as: <math display="block" id="M22"><mrow><msub><mrow><mrow><mrow><mfrac><mrow><mtext>d</mtext><mi>ρ</mi></mrow><mrow><mtext>d</mtext><mi>T</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mtext>total</mtext></mrow></msub><mo>=</mo><msub><mrow><mrow><mrow><mfrac><mrow><mtext>d</mtext><mi>ρ</mi></mrow><mrow><mtext>d</mtext><mi>T</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mtext>fuel density</mtext></mrow></msub><mo>+</mo><msub><mrow><mrow><mrow><mfrac><mrow><mtext>d</mtext><mi>ρ</mi></mrow><mrow><mtext>d</mtext><mi>T</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mtext>doppler</mtext></mrow></msub><mo>+</mo><msub><mrow><mrow><mrow><mfrac><mrow><mtext>d</mtext><mi>ρ</mi></mrow><mrow><mtext>d</mtext><mi>T</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mtext>graphite</mtext></mrow></msub><mo> </mo><mo>.</mo></mrow></math> The fuel density reactivity coefficient is the change in the value of the core reactivity when the density of molten salt is changed via a one-unit rise in temperature and other conditions remain the same (including the temperature of the molten salt). For 2/4/6/8/10 mol% HN, the density of the molten salt changes from 2.195/2.439/2.665/2.873/3.068 g/cm 3 to 2.156/2.397/2.619/2.825/3.017 g/cm 3 when the temperature is changed from 900 K to 1000 K. The fuel Doppler coefficient and the GTRC are the changes in the values of the core reactivity per unit change in the molten salt and graphite, respectively, with all other conditions remaining the same. Each part of the total TRC can be analyzed with Eq. (9) and is discussed separately below. Based on the smTMSR model, the reliability of the four-factor method can be verified through comparing the TRCs calculated by Eq. (9) and Eq. (7). The differences are shown in Table 2 Table 2 . 1 1 HN (mol%) 5 1 VF (%) 1 1 1 1 5 1 1 10 1 1 15 1 1 20 1 1 25 1 1 2 1 1 0.23 1 1 -0.33 1 1 0.01 1 1 -0.08 1 1 -0.11 1 1 4 1 1 0.12 1 1 -0.17 1 1 -0.02 1 1 -0.12 1 1 -0.06 1 1 6 1 1 -0.10 1 1 -0.14 1 1 -0.04 1 1 0.08 1 1 -0.23 1 1 8 1 1 -0.03 1 1 -0.07 1 1 -0.01 1 1 -0.08 1 1 0.04 1 1 10 1 1 0.03 1 1 -0.14 1 1 -0.17 1 1 -0.21 1 1 -0.29 The statistical errors in the TRC calculated by the TRC definition and the four-factor formula method are <math id="M23"><mo>±</mo></math> 0.12 pcm/K and <math id="M24"><mo>±</mo></math> 0.31 pcm/K, respectively. According to Table 2 Table 2 , the differences are within the scope of the statistical error, and the calculation results were basically identical, proving that the four-factor method in the current work is reliable. 4. Results and discussion 1 4.1 Effective multiplication factor 2 Fig. 2 Fig. 2 shows the change in k eff in the 25 different cases mentioned above, with the HN range 2 mol%–12 mol% and the VF range 5%–25%. It can be seen that k eff increases first and then decreases with increase in the VF. The gradient of k eff as the heavy nuclei amount is decreased is significantly larger. The maximal k eff , which shows the highest fuel utilization, is located at 10% VF and 12 mol% HN. The dashed line connects the turning points of the contour lines in the direction of the VF. It indicates that k eff increases as VF increases on the left side of the dashed line, and that k eff decreases as VF decreases on the right side of the dashed line. Thus, the dashed line in Fig. 2 Fig. 2 represents the critical points of the overmoderated region and the undermoderated region. Compared with the general MSBR, the critical points tend to occur at smaller VF [ 9 9 ]. Different kinds of driver fuel will lead to different locations of the critical points between the undermoderated region and the overmoderated region. For example, the fuel composition of the TMSR, which is a thorium-based MSBR [ 9 9 ], is 71.7 LiF+16 BeF 2 +12.3 (ThF 4 + 233 UF 4 ), different from that of the smTMSR. The critical point is located at the point where the VF is about 17% and the HN is 12.3 mol% in the TMSR [ 9 9 ], although the VF of the smTMSR is near 10% with 12 mol% HN. The reason for the different critical points may be that the fission cross section of 235 U is larger than that of 233 U. 4.2 Fuel density coefficient 2 The fuel density coefficient is positively correlated with the heavy nuclei amount (VF and HN), as shown in Fig. 3 Fig. 3 . The absolute value of the negative coefficient decreases with increasing the heavy nuclei amount and then changes from negative to positive. The critical point curve of the fuel density temperature coefficient is almost the same as the critical point curve of the overmoderated region and the undermoderated region, as shown in Fig. 2 Fig. 2 . This is because the effect of the decrease in the fuel salt on the reactivity is consistent with the VF decrease. As analyzed in subsection 4.1, this implies that the VF of the smTMSR should be smaller than that of the TMSR in order to obtain a negative density effect. The fuel density coefficient was further analyzed by the four-factor method, as presented in Fig. 4 Fig. 4 . The resonance escape coefficient has a significantly positive effect on the TRC in the undermoderated region; however, the thermal utilization coefficient and neutron leakage have an obviously negative influence on the TRC in the overmoderated region. In Fig. 5 Fig. 5 and Fig. 6 Fig. 6 , two conditions (VF=5%, HN=2 mol% and VF=20%, HN=12 mol%) were chosen. The former condition is in the overmoderated region and the latter is in the undermoderated region. As observed in Fig. 5 Fig. 5 , the resonance escape coefficient is mainly affected by 232 Th. Because of the decrease in the fuel salt density, the ratio of moderation to fuel will increase and the neutron spectrum will be softer ( Fig. 6 Fig. 6 ). More fast neutrons are moderated by graphite instead of absorbed by 232 Th in the resonance region. The resonance absorption of 232 Th causes an obvious decrease in the undermoderated region when the fuel salt density decreases, so that the resonance escape coefficient is positive and will be more obvious in the undermoderated region. The thermal utilization coefficient is mainly caused by neutron absorption by the graphite, as shown in Fig. 5 Fig. 5 . When the fuel salt density decreases and the ratio of moderation to fuel increases, the neutron absorption by graphite increases, especially in the overmoderated region. Thus, the thermal utilization coefficient is negative. In the undermoderated region, the thermal neutron absorption increases because the neutron resonance escape probability increases. However, as the increase in graphite absorption decreases, so does the value of the thermal utilization coefficient. Neutron leakage is similar to graphite absorption; therefore, the non-leakage coefficient is negative and the absolute value decreases with the increase in the heavy nuclei amount. 4.3 Fuel Doppler coefficient 2 Fig. 7 Fig. 7 presents the fuel Doppler coefficient for different VFs and HNs. The resonance escape coefficient has a significantly positive effect on the TRC in the undermoderated region; however, the thermal utilization coefficient and neutron leakage have a negative influence on the TRC in the overmoderated region. The Doppler effect [ 3 3 , 29 29 ] will broaden the resonance peaks of heavy nuclei and reduce the energy self-shielding effect because of the rise in the temperature of the fuel salt, increasing the neutron resonance absorption. Usually, fertile fuels such as 238 U and 232 Th have higher resonance peaks than fission fuels like 235 U, as shown in Fig. 10 Fig. 10 . More neutrons are absorbed by 232 Th and 238 U than by 235 U ( Fig. 9 Fig. 9 ), especially in the undermoderated region. Therefore, the fuel Doppler coefficient caused by the resonance escape coefficient is always negative, and is inversely proportional to the heavy nuclei amount, as shown in Fig. 8 (a) Fig. 8 (a) . The other main factors of the fuel Doppler coefficient come from the thermal utilization coefficient and non-leakage coefficient in the overmoderated region, as shown in Fig. 8 (b) and (c) Fig. 8 (b) and (c) . It is noteworthy that the carrier fluoride salt is also a moderator. The Maxwell peak shifts to the high energy region because the temperature increases ( Fig. 10 Fig. 10 ). The thermal scattering of the carrier salt increases, so the fission of 235 U decreases and the graphite absorption and leakage increase ( Fig. 9 Fig. 9 ). The fission cross section of 235 U decreases faster, which will make the decrease of the fission reaction more serious. The increase in the graphite absorption and the decrease in 235 U fission lead to a negative thermal utilization coefficient. The increase in leakage results in a negative non-leakage coefficient. These effects are more obvious in the overmoderated region, and therefore, the thermal utilization coefficient and the non-leakage coefficient are more negative. 4.4 Fuel temperature reactivity coefficient 2 The sum of the fuel density coefficient and the fuel Doppler coefficient are the FTRC, as shown in Fig. 11 Fig. 11 . The FTRC is positively correlated with the heavy nuclei amount (VF and HN). The absolute value of the negative coefficient decreases with the increase of the heavy nuclei amount. Compared with TMSR [ 9 9 ], the FTRC in the smTMSR with 12 mol% HN and the same VF showed little difference, as shown in Table. 3 Table. 3 . In the low HN region, the fuel density effect and the Doppler effect both lead to a deeply negative TRC. This will lead to a significant change in the TRC during the operation of the smTMSR when the HN is continuously refueled from 6 mol% (if the VF=10% and the initial k eff is close to 1) to 12 mol%. This will not happen in the TMSR because the heavy nuclei will be conservative, and always be kept at around 12% mol HN [ 9 9 ]. 1 1 Reactor 4 1 VF (%) 1 1 1 1 5 1 1 10 1 1 15 1 1 20 1 1 TMSR 1 1 -3.2 1 1 -2.5 1 1 -2 1 1 -2.3 1 1 smTMSR 1 1 -3.1 1 1 -2.5 1 1 -2.3 1 1 -2.5 4.5 Graphite temperature reactivity coefficient 2 Fig. 12 Fig. 12 presents the graphite reactivity coefficient for different VFs and HNs. The graphite temperature coefficient is always negative, and becomes more negative with lower heavy nuclei amount, which is different from the Th-U cycle in TMSR [ 9 9 ]. In the TMSR, the graphite temperature reactivity coefficient is positive and decreases with increasing VF. The thermal utilization coefficient plays a significant positive role in the overmoderated region; however, the neutron leakage has an obvious negative influence in the undermoderated region. Similar to the salt effect analyzed in subsection 4.3, the graphite thermal scattering is enhanced and the Maxwell peak shifts to the higher energy region; hence, the neutron spectrum becomes harder ( Fig. 15 Fig. 15 ). Since the fission cross section of 235 U decreases faster than the absorption cross section of 232 Th, the hardening of the energy spectrum leads to more decrease in the 235 U fission. As a result, the neutron leakage increases, which has a negative effect on the TRC, as shown in Fig. 13 (b) Fig. 13 (b) . In the TMSR, the rate of decrease in the fission cross section of 233 U is slower, so the thermal scattering of graphite has a positive contribution. However, the graphite absorption probability decreases, as shown in Fig. 14 Fig. 14 , which has a positive effect on the thermal utilization coefficient, as shown in Fig. 13 (a) Fig. 13 (a) . The neutron leakage is more obvious than the decrease in the neutron absorption by graphite. As a result, the GTRC is negative. With higher heavy nuclei amounts, the neutron leakage will decrease, and the decrease in the graphite absorption will become smaller, so the absolute value of the GTRC will decrease. 4.6 Total temperature reactivity coefficient 2 Fig. 16 Fig. 16 shows the total TRC of the smTMSR for different VFs and HNs. The total temperature coefficients are negative (-15 pcm/K to -3 pcm/K) in the range 5% to 25% VF and 2 mol% to 12 mol% HN. In Fig. 16 Fig. 16 , the main factors affecting the temperature coefficient in the undermoderated and overmoderated region are added. Only the fuel density effect has a positive effect on the TRC in the undermoderated region. All other factors make a negative contribution. The fuel Doppler coefficient mainly contributes in the undermoderated region. In addition, the thermal scattering from both the salt and graphite have a significant influence in the overmoderated region. 5. Conclusion 1 The effect of the VF and HN on the TRC of the smTMSR has been analyzed in this paper. The four-factor formula method and reaction rate method were used to determine the reasons for the changes in the TRC, including the fuel density effect, the fuel Doppler effect, and the carrier salt and graphite thermal scattering. Because the driver fuel and fuel cycle mode in the smTMSR are different from those in the TMSR, some new findings can be summarized as follows: (1) The fuel density effect in the undermoderated region is positive, and is mainly caused by the resonance escape of fertile fuel, whereas in the overmoderated region, the fuel density effect is negative because the graphite absorption and the neutron leakage play a leading role. The VF in the smTMSR should be smaller than that in the TMSR in order to obtain a negative density effect, possibly because the cross section of 235 U is larger than that of 233 U. (2) The thermal scattering effects of both salt and graphite are obviously negative in the overmoderated region. The thermal scattering effect of graphite is the main reason for the negative GTRC, whereas the contribution of this effect is positive in TMSR. This is because the fission cross section of 235 U falls faster in the thermal region than that of 233 U. 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Energy 35 ( 5 ), 904 - 916 ( 2008 ). doi: 10.1016/j.anucene.2007.09.003 http://doi.org/10.1016/j.anucene.2007.09.003 Analysis of the reactivity coefficients of the advanced high-temperature reactor for plutonium and uranium fuels 10.1007/s41365-019-0666-3 This work is supported by the Chinese TMSR Strategic Pioneer Science and Technology Project (No.XDA02010000) and the Frontier Science Key Program of Chinese Academy of Sciences (No.QYZDY-SSW-JSC016). 2018-12-27 2019-04-08 2019-05-14 2019-08-13 2019-09-01 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2019 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-019-0666-3 http://dx.doi.org/10.1007/s41365-019-0666-3 . 2019 Nuclear Science and Techniques 09 30